On expansive homeomorphism of uniform spaces

We study the notion of expansive homeomorphisms on uniform spaces. It is shown that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a Hausdor space and hence a regular space. Further, we characterize orbit expansive homeomorphisms in terms of topologically expansive homeomorphisms and conclude that if there exist a topologically expansive homeomorphism on a compact uniform space then the space is always metrizable.

Invariants of Uniform Conjugacy on Uniform Dynamical System

In this paper, we present some important dynamical concepts on uniform space such as the uniform minimal systems, uniform shadowing, and strong uniform shadowing. We explain some definitions and theorems such as definition uniform expansive, weak uniform expansive, uniform generator, and the proof of the theorems for them. We prove that if be a homeomorphism on a compact uniform space then has uniform shadowing if and only if has uniform shadowing, so if has strong uniform shadowing if and only if has strong uniform shadowing. We also show that and be two uniform homeomorphisms on compact uniform spaces and , if is a uniform conjugacy from to , then . Besides some other results.

Paracompact-type mappings

Recently a new direction of uniform topology called the uniform topology of uniformly continuous mappings has begun to develop intensively. This direction is devoted, first of all, to the extension to uniformly continuous mappings of the basic concepts and statements concerning uniform spaces. In this case a uniform space is understood as the simplest uniformly continuous mapping of this uniform space into a one-point space. The investigations carried out have revealed large uniform analogs of continuous mappings and made it possible to transfer to uniformly continuous mappings many of the main statements of the uniform topology of spaces. The method of transferring results from spaces to mappings makes it possible to generalize many results. Therefore, the problem of extending some concepts and statements concerning uniform spaces to uniformly continuous mappings is urgent. In this article, we introduce and study uniformly R-paracompact, strongly uniformly R-paracompact, and uniformly R-superparacompact mappings. In particular, we solve the problem of preserving R-paracompact (respectively, strongly uniformly R-paracompact, uniformly R-superparacompact) spaces towards the preimage under uniformly R-paracompact (respectively, strongly uniformly R-paracompact, uniformly R-superparacompact) mappings.